Zapier is an integration service. It is designed to make it easy to hook multiple applications or systems together using configurable logic rather than code. In other words, you don't necessarily need to be a programmer to be able to make the integration happen.
To make this work, each service being integrated needs some code to be able to talk to Zapier's APIs - this is either provided by Zapier, by the service/software vendor, by a third party, or by yourself if you know how.
It's a system-level tool, you the site administrator would set up your applications to communicate via Zapier to enable easier integration between them.
Typical usage scenarios including synchronising user databases between multiple applications (eg CRM system, Bookkeeping system, Helpdesk system), or perhaps sending invoices from your invoicing system to your bookkeeping system, or having your Tweets sent automatically to your Facebook page, etc.
The possibilities are almost endless, it just depends on what integrations are available.
In order for XenForo to talk to other Zapier integrated systems, someone would need to write a XenForo plugin which talks to the Zapier APIs.
